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Concept
Equation solving
Summary
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.
A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.
An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1).
The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y.
However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... to denote the known variables, which are often called parameters.
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